You ask about HIIT (High INTENSITY Interval Training), but then described Interval Training. The underlying goal of HIIT is to drive your heart rate to near maximum as quickly as possible and then allow your body to recover just enough that you can exert maximum effort again, thereby keeping yourself near maximum effort for an extended period of time. Typically HIIT is practiced by sprinting for 10-30 seconds, walking for 10-60 seconds and repeating 4 to 8 times.
Now, weight training tends to be somewhat HIIT llike in practice, especially when performing compound movements to near failure. When performing squats or deadlifts with a weight that you can just complete 5 reps with will definately increase your heart rate to near maxiumum in less than 30 seconds. Bench presses usually don't because they don't involve enough muscle mass, but some other upper body exercises like weighted pull ups come close. I think the flaw in using these as HIIT is that it is not possible to perform a second set at near maxiumum effort with out resting for an extended period of time (3-5 minutes). You could reduce the weight by 25% on each set and come close to teh effect of HIIT perhaps.
I also think that the best possible exercise for HIIT is sprinting (or runing stairs or hill sprints), because the human body is designed for upright running (from evolution to get away from danger or to get to food). All the other stuff we have invented (bikes, ellipticals, treadmills, rowers, etc.) just don't work the body in it most efficient manner and the manner it eveolved over millions of years to do.